FA-15876 / Floating-point arithmetic / Open access

Subnormals receive the normal exponent bias · case 01

Subnormals receive the normal exponent bias.

Verified by executionVariant 1 · 10 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The normal bias subtraction is applied to the reserved zero exponent.

THE FAILURE

The normal bias subtraction is applied to the reserved zero exponent.

Unsuccessful approach: Using zero exponent for tiny values produces a huge scale error.

Case contract

Decode an unsigned 64-bit IEEE binary64 encoding into sign, kind, unbiased exponent, integer significand, quiet flag and payload. Nonfinite fields use null; zero and subnormal exponents are -1022.

Why this case matters

An offline floating representation model isolates a reproducible arithmetic fault.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import struct
N = 1
observations = []
def solve(bits):
    sign = -1 if bits >> 63 else 1
    exp = (bits >> 52) & 2047
    fraction = bits & ((1 << 52)-1)
    if exp == 2047:
        kind = 'nan' if fraction else 'infinity'
        quiet = bool(fraction & (1 << 51)) if fraction else None
        payload = fraction & ((1 << 51)-1) if fraction else None
        return [sign, kind, None, None, quiet, payload]
    kind = 'zero' if exp == 0 and fraction == 0 else 'subnormal' if exp == 0 else 'normal'
    unbiased = exp-1023
    significand = fraction | (1 << 52) if exp else fraction
    return [sign, kind, unbiased, significand, None, None]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('positive zero', solve(0), [1,'zero',-1022,0,None,None])
check('negative zero', solve(1<<63), [-1,'zero',-1022,0,None,None])
check('tiny subnormal', solve(N), [1,'subnormal',-1022,N,None,None])
check('negative subnormal', solve((1<<63)|N), [-1,'subnormal',-1022,N,None,None])
check('normal exact power', solve((1023+N)<<52), [1,'normal',N,1<<52,None,None])
check('normal fraction', solve((1023<<52)|N), [1,'normal',0,(1<<52)|N,None,None])
check('positive infinity', solve(2047<<52), [1,'infinity',None,None,None,None])
check('negative infinity', solve((1<<63)|(2047<<52)), [-1,'infinity',None,None,None,None])
check('quiet payload', solve((2047<<52)|(1<<51)|N), [1,'nan',None,None,True,N])
check('signaling payload', solve((2047<<52)|N), [1,'nan',None,None,False,N])
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
positive zero	`[1, 'zero', -1023, 0, None, None]`	`[1, 'zero', -1022, 0, None, None]`	Failed
negative zero	`[-1, 'zero', -1023, 0, None, None]`	`[-1, 'zero', -1022, 0, None, None]`	Failed
tiny subnormal	`[1, 'subnormal', -1023, 1, None, None]`	`[1, 'subnormal', -1022, 1, None, None]`	Failed
negative subnormal	`[-1, 'subnormal', -1023, 1, None, None]`	`[-1, 'subnormal', -1022, 1, None, None]`	Failed
normal exact power	`[1, 'normal', 1, 4503599627370496, None, None]`	`[1, 'normal', 1, 4503599627370496, None, None]`	Passed
normal fraction	`[1, 'normal', 0, 4503599627370497, None, None]`	`[1, 'normal', 0, 4503599627370497, None, None]`	Passed
positive infinity	`[1, 'infinity', None, None, None, None]`	`[1, 'infinity', None, None, None, None]`	Passed
negative infinity	`[-1, 'infinity', None, None, None, None]`	`[-1, 'infinity', None, None, None, None]`	Passed
quiet payload	`[1, 'nan', None, None, True, 1]`	`[1, 'nan', None, None, True, 1]`	Passed
signaling payload	`[1, 'nan', None, None, False, 1]`	`[1, 'nan', None, None, False, 1]`	Passed

SHA-256 / 55d2e4d3b4008c45b7d7ed2e8da44b31ea4bcf9676df064c5b7bbfb495c8eba1

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import struct
N = 1
observations = []
def solve(bits):
    sign = -1 if bits >> 63 else 1
    exp = (bits >> 52) & 2047
    fraction = bits & ((1 << 52)-1)
    if exp == 2047:
        kind = 'nan' if fraction else 'infinity'
        quiet = bool(fraction & (1 << 51)) if fraction else None
        payload = fraction & ((1 << 51)-1) if fraction else None
        return [sign, kind, None, None, quiet, payload]
    kind = 'zero' if exp == 0 and fraction == 0 else 'subnormal' if exp == 0 else 'normal'
    unbiased = exp-1023 if exp else 0
    significand = fraction | (1 << 52) if exp else fraction
    return [sign, kind, unbiased, significand, None, None]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('positive zero', solve(0), [1,'zero',-1022,0,None,None])
check('negative zero', solve(1<<63), [-1,'zero',-1022,0,None,None])
check('tiny subnormal', solve(N), [1,'subnormal',-1022,N,None,None])
check('negative subnormal', solve((1<<63)|N), [-1,'subnormal',-1022,N,None,None])
check('normal exact power', solve((1023+N)<<52), [1,'normal',N,1<<52,None,None])
check('normal fraction', solve((1023<<52)|N), [1,'normal',0,(1<<52)|N,None,None])
check('positive infinity', solve(2047<<52), [1,'infinity',None,None,None,None])
check('negative infinity', solve((1<<63)|(2047<<52)), [-1,'infinity',None,None,None,None])
check('quiet payload', solve((2047<<52)|(1<<51)|N), [1,'nan',None,None,True,N])
check('signaling payload', solve((2047<<52)|N), [1,'nan',None,None,False,N])
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
positive zero	`[1, 'zero', 0, 0, None, None]`	`[1, 'zero', -1022, 0, None, None]`	Failed
negative zero	`[-1, 'zero', 0, 0, None, None]`	`[-1, 'zero', -1022, 0, None, None]`	Failed
tiny subnormal	`[1, 'subnormal', 0, 1, None, None]`	`[1, 'subnormal', -1022, 1, None, None]`	Failed
negative subnormal	`[-1, 'subnormal', 0, 1, None, None]`	`[-1, 'subnormal', -1022, 1, None, None]`	Failed
normal exact power	`[1, 'normal', 1, 4503599627370496, None, None]`	`[1, 'normal', 1, 4503599627370496, None, None]`	Passed
normal fraction	`[1, 'normal', 0, 4503599627370497, None, None]`	`[1, 'normal', 0, 4503599627370497, None, None]`	Passed
positive infinity	`[1, 'infinity', None, None, None, None]`	`[1, 'infinity', None, None, None, None]`	Passed
negative infinity	`[-1, 'infinity', None, None, None, None]`	`[-1, 'infinity', None, None, None, None]`	Passed
quiet payload	`[1, 'nan', None, None, True, 1]`	`[1, 'nan', None, None, True, 1]`	Passed
signaling payload	`[1, 'nan', None, None, False, 1]`	`[1, 'nan', None, None, False, 1]`	Passed

SHA-256 / f84f2547303096b9f0f8c74a5bbf79003536303e19772422d5cb90ff502299fd

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 10 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

Controlled binary64 or explicitly stipulated miniature format; no hardware exception flags or platform floating environment are modeled. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:39:31.115738+00:00.

Case digest / 8ecc07c4fe18c18f67e61d329b1f5b0d7a8033f4ca4534370f842b309a598517